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PHYS. REV. B 98, 035414 (2018)

Thermal rectification with interacting electronic channels: Exploiting degeneracy,

quantum superpositions and interference

Alejandro Marcos-Vicioso,1 Carmen Lopez-Jurado,1 Miguel Ruiz-Garcia,2, 3 and Rafael Sanchez2, 4

1Escuela Politecnica Superior, Universidad Carlos III de Madrid, 28911 Leganes, Spain2Gregorio Millan Institute for Fluid Dynamics, Nanoscience,

and Industrial Mathematics, and Department of Materials Science and Engineering,Universidad Carlos III de Madrid, 28911 Leganes, Spain

3Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA4Departamento de Fısica Teorica de la Materia Condensada,

Universidad Autonoma de Madrid, 28049 Madrid, Spain

This paper explores different mechanisms that induce thermal rectification in the nanoscale. Thepresence of interacting energy channels combined with simple asymmetries is sufficient for promotingthe desired behavior. We use simple quantum dot configurations, identifying the basic propertiesthat enhance rectification for each case: the size of a quantum dot state space (which suggests theuse of scaled up systems with many interacting channels), tunneling asymmetries due to coherenttunneling in a double quantum dot, or quantum interference in a triangular triple quantum dot. Anefficient and tunable thermal diode is proposed using a channel capacitively coupled to a mesoscopicswitch.

I. INTRODUCTION

Conventional electronics is constantly increasing com-putational power through the miniaturization of its basiccomponents. The operation of a circuit can be seriouslyharmed by the heat dissipated in its working components.Gaining control over nanoscopic heat currents is hencevital for improving the operation of electronic devices.There are different approaches to alleviate or even take

advantage of the dissipated heat in electronic devices.One possibility is to convert it into power by thermoelec-tric engines1,2. One can also think of doing useful op-erations which are only driven by heat3,4, for which oneneeds to find thermal analogs of electrical componentslike a transistor or a diode which work at the nanoscale.Research in this direction has been boosted by the recentadvances in the detection of mesoscopic heat currents5–9.Any diode, including thermal ones, require a spacial

asymmetry that affects the current propagation betweentwo terminals10. This has led to proposals of thermalrectifiers based on broken mirror symmetry that use aseries of systems with different spectral densities. Recentexamples include linear lattices11, superconducting junc-tions12–15, normal-superconducting junctions16, metal-lic islands17, quantum Hall tunnel barriers18, metal-dielectric interfaces19, qubits20,21, or resonators22. Otherpossiblities include energy-dependent couplings23 or theasymmetric coupling to a third reservoir acting as an en-vironment with which the system exchanges energy24–28.One can also use the effect of electron-electron interac-

tions. In quantum dot systems with discrete spectral den-sities, they are responsible for strong nonlinearities29–36.Several quantum dots can be coherently coupled to formdifferent configurations, which enables one to locally con-trol the density of states. For instance, the symmetry ofquantum superpositions can be controlled by gate volt-ages in linearly-coupled double37,38 or triple quantum

dots39. More complicated spacial arrangements40–42 in-troduce different tunneling paths which give rise to quan-tum interference effects43–47, under the appropriate sym-metries48.

In this paper we will restrict ourselves to the study ofsimple configurations of quantum dots, see Fig. 1. Withthis minimalistic approach we aim to reduce the num-ber of degrees of freedom helping us to identify the rele-vant processes. In all the considered configurations trans-

PSfrag replacements12

34

(a)

(b)

(c)

L R

A

B

C

ΓL ΓR

TL TR

τLR

τAC

τBC

FIG. 1. Scheme of the different quantum dot configurationsdiscussed in the paper. (a) A single quantum dot is coupledto two terminals with asymmetric tunneling rates Γl. Thereare two possible states the electrons can tunnel to. (b) Adouble quantum dot in series. Hybridization of the quantumdot orbitals due to coherent tunneling, τLR, introduces L-Rasymmetric and energy-dependent tunneling rates even if theleft and right barriers are identical. (c) A triple quantum dotin a triangular configuration introduces directionality due tothe interference of trajectories coming from the left lead.

2

port occurs via channels that are correlated via strongCoulomb interactions. We assume the Coulomb block-ade regime, where the system can be occupied by up toone electron. The mechanism which breaks mirror sym-metry and enables rectification is different in each case.In the simplest case, a single quantum dot is tunnel-

coupled to two terminals, the quantum dot levels needto be asymmetrically coupled to the left and right leads,but remarkably no energy-dependent tunneling is needed,cf. Fig. 1(a). Indeed, we find the surprising resultthat the multiplicity of the quantum dot states increasesthe rectification. The left-right asymmetry and energy-dependence of the tunneling rates can be manipulatedby controlling the tunneling hybridization in a doublequantum dot, cf. Fig. 1(b), which helps to increase therectification coefficient. More drastically, a spacially-asymmetric composition with two quantum dots coupledto the left lead and only one coupled to the right can workas a rectifier even if the tunnelig barriers are all identi-cal, cf. Fig. 1(c). The effect is in this case due to theformation of superpositions of the quantum dots coupledto the left which avoid the occupation of the remain-ing quantum dot49–52. They form a blocking channel inparallel to those that support the current. This prop-erty can be then applied to simpler configurations withtwo capacitively coupled quantum dots in parallel53,54,where fluctuations in one quantum dot affect the currentthrough the other one 55,56. This process is reminiscent ofdynamical channel blockade57,58 and achieves huge recti-fication coefficients at configurations with maximal heatcurrents.The paper is organized as follows. In Sec. II we present

the general formalism which is applied to the differentconfigurations. The effect of the dimensionality of thestate space is analyzed in Sec. III, and asymmetries aris-ing from coherent tunneling are introduced in Sec. IV.A triple quantum dot where a superposition of statesis only coupled to one of the terminals is presented inSec. V, whereas a similar effect is used in a simpler con-figuration with two quantum dots in Sec. VI. Conclusionsare discussed in Sec. VII.

II. MODEL AND EQUATIONS

Along this work we will show how different configura-tions of quantum dots can be tuned to exhibit a thermal-diode behavior. In particular, we will consider systemswith one, two, and three quantum dots connected to tworeservoirs at different temperatures, cf. Fig. 1. We fur-thermore assume that the electrostatic charging energyof any of these systems is sufficiently large that the totalnumber of electrons does not exceed one.For the systems appearing in Fig. 1 we are only inter-

ested on the stationary thermal currents. The informa-tion of the occupation of every energy level, accountingalso for coherences between them, is contained in thedensity matrix ρ. Its evolution follows a master equation

d

dtρ = −

i

h[Hs, ρ] +

∑

l,X,α

D[LlXα, ρ], (1)

where the index l=L,R accounts for the left and rightleads, and α = ± refers to tunneling in/out of the system.The first term on the right hand side of Eq. (1) accountsfor the coherent evolution of the isolated system. Thesecond term introduces the tunneling between the systemand the reservoirs, which will be specified in terms of therelevant states X of each configuration. We assume aBorn-Markov approximation, valid in the weak tunnelingregime, ΓlX ≪ kBT

59, see below. The dissipator is givenby the usual Lindblad form:

D[L, ρ] = LρL† −1

2

{

L†L, ρ}

+. (2)

In particular,

LlX+ =√

Γ+lX |X〉〈0| (3)

LlX− =√

Γ−lX |0〉〈X |, (4)

where the rates for tunneling in/out are: Γ+lX =

ΓlXf(EX−µl, Tl), and Γ−lX = ΓlX [1 − f(EX−µl, Tl)],

with the Fermi function f(E, T ) = [1 + exp(E/kBT )]−1

giving the electronic distribution of a lead at temperatureT . The transparency of barrier l, ΓlX , may depend onthe energy level involved in the transition, X . We assumea wide band approximation such that any X-dependenceof the tunneling transparency is only due to the inter-nal spectrum of the quantum dot system. Finally, theequilibrium Fermi energy has been set to zero for conve-nience. In most cases, we will assume that there is noelectric potential applied to the reservoirs, i.e. µl = 0,except when explicitly mentioned.With the stationary solution of the master equation

(1) for the density matrix elements, ρij = 0, we obtainthe dc charge and heat currents:

Il = e∑

X

(ρ00Γ+lX − ρXXΓ−

lX) (5)

Jl =∑

X

(EX − µl)(ρ00Γ+lX − ρXXΓ−

lX), (6)

respectively. As we are interested in the rectification ofheat flows, we need to compute the heat current throughthe system in response to opposite temperature gradi-ents.There are two possible configurations to be considered:

(i) short circuit, where eV = µL − µR = 0, and (ii) opencircuit, where a (thermo)voltage develops to the condi-tion Il = 0. This paper will mostly focus on case (i),although case (ii) will be discussed for some configura-tions. In both cases, no power is generated in the sys-tem, such that heat is conserved: JL + JR = 0. Thereis hence no ambiguity in defining the forward and back-ward responses in a single terminal, e.g. J+=JL(TL=T +∆T, TR=T ) and J−=−JL(TL=T, TR=T+∆T ), depend-ing on what terminal has a temperature increase ∆T .With these, we define the rectification coefficient:

R =

∣

∣

∣

∣

J+ − J−

J+ + J−

∣

∣

∣

∣

, (7)

3

PSfrag replacements

1234

(a)(b)(c)

LRAB

∆EΓL1 ΓR1

ΓL2 ΓR2

TL

TR

τLRτAC

τBC

FIG. 2. Two-level quantum dot coupled to two terminals.Left-right asymmetric and level-dependent tunneling lead tofour different tunneling rates, Γli. The latest can be due toan energy splitting ∆E in energy-dependent barriers, or to anadditional degree of freedom (e.g. spin).

which is bounded between 0 (no rectification) and 1 (anideal thermal diode).

III. DEGENERACY IN A QUANTUM DOT

Let us first consider the case of a single quantumdot60,61, see Fig. 1(a). It is important to emphasize thatmirror symmetry breaking is not sufficient to producerectification. To show this, it is useful to first explorea simple model with only one available energy state εin the quantum dot. This is the case if one can neglectthe effect of spin (for instance, if the leads are fully spin-polarized). The coupling of this state to the left andright leads is parametrized by the tunneling rates Γl1.The heat current (at the condition of no voltage bias) isgiven by:

J1(ε) = εΓL1ΓR1

ΓL1+ΓR1

[f(ε, TL)− f(ε, TR)], (8)

as shown in App. A. As the temperature dependence onlyenters in the difference of Fermi functions, the current isantisymmetric under the change TL ↔ TR, leading toJ+ = J−, i.e. no rectification. Note also that havingadditional states does not change the situation providedthat they do not interact with each other, as discussedin App. B.However, the behavior dramatically changes when con-

sidering several channels that are correlated via inter-actions. We are interested here in the simplest caseof two states that exclude each other (due to strongCoulomb blockade), as depicted in Fig. 2. We label them|X〉 = |1〉, |2〉 for simplicity. In our case, they can corre-spond to the two possible spin states of the electron thatoccupies a single-level quantum dot, which can be splitby ∆E, e.g. due to an applied magnetic field. We re-mark here that the interplay of two states (a ground andan excited state) was used to interpret the rectificationof a quantum dot in experiments29.As there are no internal dynamics in the quantum dot,

the first term on the right-hand side of Eq. (1) does notcontribute to the evolution of the system. Hence, off-diagonal elements of the density matrix are uncoupled

0

2

4

6

8

10

−10 −5 0 5 10

0

2

4

6

8

10

0

0.4

0

0.02

-10 0 10

3

∆E/kBT

ε/kBT

0

0.02(c)

R

∆E/kBT

0

0.3

(a)

J+[ΓkBT ]

J±

[ΓkBT] J+

J−

(b)

R

ε/kBT

04

(d) ∆E/kBT :

FIG. 3. Rectification of a single quantum dot whose spinstates are split by a magnetic field: ε1 = ε − ∆E/2, ε2 =ε+∆E/2, with ∆E = gµBz. (a) Heat current for ∆T = T/2,V = 0, ΓL = 2ΓR = 0.2kBT , with Γ = ΓLΓR/ΓΣ. (b) Forwardand backward currents for Bz = 0. (c) Rectification coefficientwith (d) cuts at zero and finite magnetic field. The inset in (b)zooms in the difference of the two currents around ε = 3kBT .Note that rectification is present even if Bz = 0.

from the occupations ρii and need not be taken into ac-count. The master equation then simply reads:

ρii = Γ+Σiρ00 − Γ−

Σiρii, i=1, 2 (9)

and is complemented with the normalization conditionρ00 + ρ11 + ρ22 = 1. Here we use the notation Γ±

Σi =∑

l Γ±li , with l=L,R.

The steady-state solution of Eq. (9) reads: ρ00 =Γ−Σ1Γ

−Σ2/Λ, ρ11 = Γ+

Σ1Γ−Σ2/Λ, and ρ22 = Γ−

Σ1Γ+Σ2/Λ. The

common parameter in the denominator accounts for nor-malization:

Λ = Γ−Σ1Γ

−Σ2 + Γ+

Σ1Γ−Σ2 + Γ−

Σ1Γ+Σ2. (10)

Importantly, it introduces a temperature-dependent pref-actor in the expression for the current, see Eq. (6):

JL = Λ−1∑

i

εi(

Γ+LiΓ

−Σ1Γ

−Σ2 − Γ−

LiΓ+ΣiΓ

−Σi

)

, (11)

where i = 2 for i=1, and viceversa.With this expression, we can check what the neces-

sary asymmetries are to find a finite rectification. Forinstance, it is easy to verify that mirror symmetry needsto be broken: if otherwise Γli = Γi, i.e., if tunneling ratesonly depend on the quantum dot level, we find J+ = J−.Let us consider the simplest case with energy-

independent rates, Γli = Γl, ∀i. It is maybe the mostaccessible case for experiments. One way to tune the en-ergy difference of two levels is to introduce a magneticfield Bz that induces a Zeeman splitting between the

4

states with opposite z-component, ∆E = gµBBz, where gis the gyromagnetic factor, and µB is the Bohr magneton.In this case, ε1 = ε − ∆E/2, and ε2 = ε + ∆E/2. Thebare position of the level, ε, can be tuned with a plungergate. We can clearly distinguish two regimes in Fig. 3,depending on whether the Zeeman splitting is smaller orlarger than kBT : For ∆E < kBT , both levels are withinthe window of thermal excitations and the heat currentvanishes close to the symmetric point ε = 0. Addition-ally, at ε/∆E ≈ 1/2 and −3/2, we find that J+ = J−.Otherwise, a small but finite rectification appears.In the regime ∆E ≫ kBT , charge fluctuations affect

only one state (at most). When ε > 0, the upper stateis empty and the system behaves as a single-state quan-tum dot (discussed at the beginning of this section). Inthis region, we find a sizable heat current with suppresedrectification (as expected). For ε < 0, the lower stateis occupied and blocks any transport through the otherone, a mechanism related to dynamical channel block-ade57,58. It leads to the suppression of transport, so wefind a maximal rectification coefficient of tiny heat cur-rents.Higher rectification coefficients are obtained for the

open-circuit case, discussed in Appendix C.

A. Degenerate levels

A particularly interesting case of discussion is when thetwo states have the same energy, ε1 = ε2 = ε. One couldnaıvely expect that this configuration presents no rec-tification, in analogy with the single state case, Eq. (8).However, due to the Coulomb blockade events at differentenergy levels are correlated, as tunneling into each iso-lated state is conditioned on the other one being empty.The occupation of each state, in turn, depends on bothtunneling transparencies and on temperature, and henceare different depending on which lead is hot, in general.If the tunneling rates are state-independent, Γli = Γl,

this configuration can be mapped to the single-state oneby replacing Γ+

l → 2Γ+l . We emphasize that this only

affects the tunneling-in rates: while the empty quantumdot has two states that can be occupied, there is only onepossible final state when the quantum dot is occupied.The resulting heat current (for V = 0) reads:

J2 =2εΓLΓR

∑

l Γl[1 + f(ε, Tl)][f(ε, TL)− f(ε, TR)]. (12)

Note that the denominator of the prefactor now dependson the temperature of the leads. This leads to a finitethermal rectification, which for small temperature gradi-ents can be written as:

J+2 −J−

2 =2Γαx3kB∆T 2

T (3 + 3 coshx− sinhx)2+O

(

∆T

T

)3

, (13)

with x = ε/kBT and Γ = ΓLΓR/ΓΣ. Note that it re-lies on a finite tunneling asymmetry α = (ΓL − ΓR)/ΓΣ.With this result, one immediately finds that the leading

PSfrag replacements

1234

(a) (b)(c)

ΓL

ΓR

TL

TR

τLRτAC

τBC

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20

(c)

RN/α

ε/kBT

N :12510

100∞

FIG. 4. Interacting channels in parallel realized in (a) twocapacitively coupled quantum dots, and (b) a system of self-assembled quantum dots. (c) Rectification coefficient for asystem of N capacitively coupled quantum dots in parallel,with ∆T = T/2.

order contribution of the rectification coefficient increaseslinearly with ∆T :

R2 =

∣

∣

∣

∣

∣

∣

εα∆T

2kBT 2

(

3 + 3 cosh εkBT − sinh ε

kBT

)

∣

∣

∣

∣

∣

∣

, (14)

for small gradients.The effect of the degeneracy of a quantum dot due

to spin on the tunneling rates can be explicitly detectedin an experiment62. It can be modulated by addition-ally selecting the spin of the injected currents, e.g. withferromagnetic contacts. A quantum dot coupled to fully-polarized ferromagnetic contacts would recover the re-sults of a single state in Eq. (8), see App. A. Controllingthe polarization of the leads would then switch the rec-tification on.

B. Scaling the rectification up

This approach opens the possibility of enhancing therectification by using a larger number of quantum dots.In this way, the number of accessible states increases, andso does magnitude of the total heat current. Considerfor instance an array of N capacitively-coupled quan-tum dots which are connected to the same two terminals(see Figs. 4(a) and 4(b) for possible setups with N=2,or larger). The occupation of one of them increases thecharging energy of its neightbours by the Coulomb inter-action. Assuming that this energy is large (compared tothe energy of thermal fluctuations, kBTl), the equationsfor the new system can also be obtained from the single-state case by replacing Γ+

l → 2NΓ+l (2 is for spin). This

5PSfrag replacements

L R

∆E

|α−|2ΓL |β−|

2ΓR

|α+|2ΓL

|β+|2ΓR

TL

TR

τLR

τAC

τBC

FIG. 5. Double quantum dot in series. Electron delocalizationdue to coherent interdot tunneling, τLR, leads to hybridizationof the quantum dot states. The relative weight of the orbitalsin each quantum dot affects the tunneling rates.

is the opposite case to the one with many non-interactingchannels discussed in App. B.The rectification in this case reads:

RN =

∣

∣

∣

∣

α(2N−1)[f(ε, T+∆T )−f(ε, T )]

{2 + (2N−1)[f(ε, T+∆T )+f(ε, T )]}

∣

∣

∣

∣

, (15)

which is plotted in Fig. 4(c). For every given configura-tion (fixed rates, temperatures, and level position), RN

increases with N . Of course, a configuration with a largenumber of quantum dots which are all so strongly cou-pled that there can only be one electron in all of them isquite unrealistic. It, however, motivates the investigationof thermal effects in densely self-assembled quantum dotlayers (which by construction include left-right asymme-tries quite naturally) or related multiplexed devices.

IV. COHERENT TUNNELING IN A DOUBLE

QUANTUM DOT

Some room for improvement is expected for systemswith combined mirror asymmetry and energy-resolvedtunneling rates. Energy-dependent asymmetries of thetunneling rates in a single quantum dot can be tuned tosome extent56, but they are usually small and difficultto control. To find the desired asymmetry, we consider adouble quantum dot coupled in series to the two termi-nals, as sketched in Fig. 1(b). Thermoelectric propertiesof this system have been measured38,63. For the sake ofsimplicity, and in order to isolate the particular effectof this geometry, we will neglect the spin degree of free-dom, even when it helps to increase the rectification, asshown in the previous section. The case with spin degen-eracy can be recovered again by doubling the tunneling-inrates.The Hamiltonian of the system takes the form37:

HDQD =∑

l=L,R

εlnl − τLR

(

c†RcL +H.c.)

, (16)

where εl is the energy of the level of each quantum dot,l=L,R, and nl its occupation operator. Coherent interdottunneling, τLR, produces the hybridization of the states|L〉 and |R〉, which leads to the formation of molecular-like orbitals:

|±〉 = α±|L〉 − β±|R〉. (17)

−10

−5

0

5

10

−10 −5 0 5 10

−10

−5

0

5

10

0

0.1

0

0.2

-10 0 10

ε R/kBT

εL/kBT

0

0.2

0.4(c)

R

ε R/kBT

0

0.1

(a)

J+[ΓkBT ]

J±

[ΓkBT]

J+

J−

(b)

R

(εL + εR)/2kBT

135

(d)

(εL−εR)/kBT :

FIG. 6. Rectification of a double quantum dot whose energylevels, εL and εR are tuned by gate voltages. (a) Heat currentfor ∆T = T/2, V = 0, τLR = kBT , ΓL = ΓR = 0.2kBT ,with Γ = ΓLΓR/ΓΣ. White-dashed lines mark the zeros ofthe eigenenergies E±. (b) Forward and backward currents forεR − εL = kBT . (c) Rectification coefficient with (d) cutsat different level splittings. It vanishes when the system issymmetric (εL = εR), and close to the conditions E± ≈ 0.

The coefficients α± = g(2τLR/[εL − εR ± ∆E]) andβ± = g([εL − εR ±∆E]/2τLR), with g(x) = (1 + x2)−1/2

and ∆E =√

(εL − εR)2 + 4τ2LR come out of the di-agonalization of the Hamiltonian (16), also giving theeigenenergies: E± = (εL + εR ±∆E)/2. When εL 6= εR,the distribution of an electron in one of the eigenstates isnot homogeneous for the two dots, as sketched in Fig. 5.In the regime τLR ≫ Γ, the dynamics is dominated

by the eigenstates. In the basis |X〉 = |0〉, |±〉, the mas-ter equation is equivalent to Eq. (9) with i = ±. Thetunneling rates from the reservoirs to the eigenstates aredetermined by the projection of the eigenstates on thelocalized basis, ΓL± = |α±|

2ΓL and ΓR± = |β±|2ΓR.

This way, hybridization effectively introduces mirror-asymmetric and energy-dependent tunneling rates, evenif the barriers are energy-independent and left-right sym-metric (ΓL = ΓR). Note that, in this case, the rates aresymmetric by pairs: ΓL± = ΓR∓. Furthermore, theseasymmetries can be experimentally tuned by controllingthe splitting εL − εR with gate voltages37,38.In the limit when the detuning between the two dots

is large, each eigenstate recovers the state of a differentquantum dot, which is coupled to a single reservoir. If, forinstance, εR ≫ εL + τLR, we have ΓL− ≈ ΓL and ΓR+ ≈ΓR, with vanishing ΓL+, and ΓR−. Hence, transport issuppressed, as shown in Fig. 6(a).On the other hand, the heat current is maximal around

the resonance condition εL = εR, where |α±| = |β±|, cf.Fig. 6. At this condition, the separation of the two levelsis minimal and given by ∆E = E+ − E− = 2|τLR|. Notealso that in this case the system is completely symmetric,

6

−10

−5

0

5

10

−10 −5 0 5 10−10

−5

0

5

10

−10 −5 0 5 10

ε R/kBT

εL/kBT

0

0.01

(b)

R

ε R/kBT

εL/kBT

0

0.04(a)

J+[ΓkBT ]

FIG. 7. Rectification of a double quantum dot in the open-circuit configuration. (a) Heat current and (b) rectificationcoefficient as functions of the position of the levels, ε1 and ε2.The same parameters as in Fig. 6 are considered.

with: ΓL± = ΓR±, resulting in R = 0.

The current becomes asymmetric as a function of en-ergy due to the Coulomb interaction. The occupationof the lowest energy level suppresses transport throughthe other, and hence current is reduced when E− < 0,see Figs. 6(a) and 6(b). If both levels are over the Fermienergy, there is no effective channel blockade. This asym-metry is clearer in the rectification coefficient, which van-ishes when the two levels are over the Fermi energy. Therectification rapidly increases when the two orbitals arewell below the chemical potential, E± ≪ 0, cf. Figs. 6(c)and 6(d). It can in principle be arbitrarily close to theoptimal value R = 1. Unfortunately, currents in thisregion are strongly suppressed and difficult to detect.

A. Open circuit configuration

The open circuit configuration is interesting by anal-ogy with a purely thermal conductor. Only heat currentsflow through the system. The left and right terminalsare floating such that a voltage Vth develops to satisfythe condition where charge current is zero. This is thethermovoltage appearing in thermoelectric engines1. Ithas to be obtained self-consistently for each configura-tion by solving the equation I(Vth) = 0. We assume forsimplicity that the voltage is symmetrically developed inthe two leads, such that µL = −eVth/2 and µR = eVth/2.

In the open circuit configuration, the heat currentshows a single peak when both levels are around theFermi energy, see Fig. 7. The developed voltage sup-presses the double-peak structure visible in Fig. 6. Onlyat ε1 = ε2 = 0, the two orbitals are symmetrically cou-pled to the leads at E± = ±∆E/2, such that I = 0 andthe two cases (open-circuit and short-circuit) coincide.This surprising effect can be understood because whenthe two channels are over the Fermi energy, only the onewith the lowest energy contributes to transport. The sys-tem effectively behaves as a single-channel, whose chargeand heat currents become proportional. As charge cur-rent is zero, also heat vanishes.

Notably, the rectification is maximal in the regionwhere the heat current peaks, except at the conditionε1 = ε2. As discussed above, the tunneling rates are

mirror-symmetric at this condition and there is no recti-fication.We emphasize the difference from the behaviour of a

two-state quantum dot (as discussed in Sec. III), dis-cussed in open circuit conditions in the App. C. There,the rectification increases in the region where transportis strongly suppressed.

V. INTERFERENCE IN A TRIPLE QUANTUM

DOT

In the previous section, we saw that tunneling asymme-tries can be introduced via the hybridization of quantumstates due to coherent tunneling. However, the largestrectification coefficients occur at conditions where theforward and backward currents are both small. In thissection we propose how to enhance these currents furtherby exploiting the effect of coherence, and introducing asetup where left to right trajectories are affected by in-terference, while right to left ones are not.This is the case of a triple quantum dot in a triangular

geometry, as pictured in Fig. 1(c). Dots A and B are con-nected to the left lead and tunnel-coupled to dot C, whichis in turn connected to the right lead. The Hamiltoniantakes the form,

HTQD =∑

l

εlnl −∑

i=A,B

τiC

(

c†Cci +H.c.)

. (18)

For simplicity, we assume that A and B are only capac-itively coupled, τAB = 0, and τAC = τBC = τ . As weare interested in left-right asymmetries, we will furtherassume that εA = εB = εAB, and that the tunneling bar-riers between the leads and all the three dots are equal.The eigenstates of this system are:

|1〉 =(

1 + x2+

)−1/2[x+(|A〉+ |B〉) + |C〉] (19)

|2〉 = 2−1/2(|A〉 − |B〉) (20)

|3〉 =(

1 + x2−

)−1/2[x−(|A〉+ |B〉) + |C〉], (21)

where x± = (εAB − εC ± ∆E31)/2τ and ∆E31 = E3 −

E1 =√

(εAB − εC)2 + 8τ2. The eigenenergies read:E1 = (εAB + εC − ∆E31)/2, E2 = εAB, and E3 =(εAB + εC +∆E31)/2.Note that quantum dot C does not contribute to eigen-

state |2〉. Even if A and B are both coupled to C, tun-neling is canceled for this particular superposition dueto destructive quantum interference. We call it a darkstate44, in analogy with quantum optics64. This is a cru-cial point: an electron tunneling from the left lead canin principle enter any of the three eigenstates. Being ei-ther in state |1〉 or in |3〉, the electron has some finiteprobability to populate quantum dot C, and therefore tosubsequently tunnel out to the right lead and contributeto transport. Differently, if the electron enters state |2〉,it will block the current (by avoiding any other state tobe occupied) until it eventually tunnels back to the leftlead. Once this happens (and before it is occupied again),transport is restored.

7PSfrag replacements

A, B C

∆EΓL1

ΓR1

ΓL2

ΓR2

TL

TR

τLRτAC

τBC

FIG. 8. Orbitals in a triple quantum dot. When the lev-els in dots A and B are degenerate, a dark state is formedwith forbidden tunneling to the level in dot C. Therefore it isuncoupled from the right lead.

On the other hand, electrons tunneling from the rightlead can only enter two states, |1〉 or |3〉, and then thereis always a finite probability that it contributes to trans-port to the left lead. There is no such destructive in-terference for electrons tunneling from the right. Hence,the asymmetry in the spacial arrangement of quantumdots translates into left and right moving electrons beingaffected by very different processes.This is reflected in the tunneling rates: ΓLi = (|αiA|

2+

|αiB|2)ΓL, and ΓRi = |βi|

2ΓR, for i=1,3, with αij = 〈i|j〉

and βi = 〈i|C〉. For the dark state we have ΓL2 = ΓL

and ΓR2 = 0. They are illustrated in Fig. 8. In thelimit τ ≫ Γl, the master equation for the states |X〉 =|0〉, |1〉, |2〉, |3〉 reads:

ρii = Γ+Σiρ00 − Γ−

Σiρii, i=1, 3 (22)

ρ22 = Γ+L2ρ00 − Γ−

L2ρ22, (23)

now taking into account the normalization ρ00 + ρ11 +ρ22 + ρ33 = 1.The resulting currents are plotted in Fig. 9 as the po-

sition of the levels εAB and εC are swept. For positiveenergies, εAB, εC > 0, small differences between J+

TQD

and J−TQD are mostly attributed to the asymmetries in

the tunneling rates due to coherent interdot tunneling,similarly to the effect discussed in Sec. IV.Most interestingly, we find a large difference when

εAB < 0. In this region, the dark state is below the chem-ical potential so it can be populated from the left lead,thus blocking the transport. If εAB < kBTL, the prob-ability that an electron in state |2〉 tunnels back to thelead is exponentially suppressed. Hence, the dynamicalblockade can not be lifted. Clearly the crossover to thissituation occurs at smaller energies (in absolute value)when the left lead is the cold one. This way, the back-ward current vanishes, while the forward current can stillincrease due to the onset of transport through state |3〉.As the temperature gradient is increased, the contribu-

tion of state |3〉 to the forward current, J+, increases. Itappears as an additional peak when E3 < 0, see Fig. 10.On the contrary, this signal is not present in the backwardcurrent. The onset of the dark state blocking is indepen-dent of the temperature of the hot lead and avoids theoccupation of |3〉. This is indeed the desired diode effect:

−10

−5

0

5

10

−10 −5 0 5 10

−10

−5

0

5

10

−10

−5

0

5

10

-10 -5 0 5 10

0

0.3

0

1

-10 0 10

ε C/kBT

εAB/kBT

0

0.3

0.6

0.9(e)

Rε C

/kBT

0

0.2

(a)

J+[ΓkBT ]

ε C/kBT

εAB/kBT

(b)

J−[ΓkBT ]

J±

[ΓkBT]

J+

J−

(c)

R

(εAB + εC)/2kBT

(d)

FIG. 9. Rectification of a triple quantum dot whose energylevels, εAB and εC are tuned by gate voltages. (a) Forwardand (b) backward heat currents for ∆T=T , V=0, τ=kBT ,ΓL=ΓR=0.2kBT , with Γ=ΓLΓR/ΓΣ. (c) Forward and back-ward currents for εAB−εC=0. (d) Cut along εAB − εC=0 ofthe rectification coefficient plotted in (e) for different levelpositions. When εAB<0, the occupation of the dark statesuppresses the backward current and high rectification coeffi-cients are attained. White-dashed lines in (a) mark the zerosof the eigenenergies Ei.

the forward current has a peak where the backward cur-rent vanishes. The rectification coefficient is then R ≈ 1for a measurable heat current.

VI. COUPLED QUANTUM DOTS

We can extend the effect shown in the last section,where a high rectification coefficient was produced bythe dynamical channel blockade, to get large rectifica-tions for simpler systems. The strong Coulomb interac-tion converted the charging/uncharging of the dark statein a switch for the current through the rest of the sys-tem. In this section, we present a minimal configurationin which this mechanism is present. It consists of twoquantum dots which are capacitively coupled, as sketchedin Fig. 11. The coupling is strong enough to avoid twoelectrons in the system. This system can be realized insemiconductor two-dimensional electron gases53–56,65,66,graphene heterostructures67, metallic islands68, couplednanowires69, or corner states in nanowire field-effect tran-

8

0

1

-10 0 10

(a)

0

1

-10 0 10

(b) ∆T/T :J±[k

BTΓ]

εAB/kBT

J+

J−

R

εAB/kBT

124

FIG. 10. Effect of temperature in a triple quantum dot. (a)Heat currents and (b) rectification coefficient for increasingtemperature gradients. We assume εAB = εC V = 0, τ =2kBT , ΓL = ΓR = 0.04kBT , with Γ = ΓLΓR/ΓΣ. Forward(backward) currents are plotted with dashed (solid) lines.

FIG. 11. Two capacitively-coupled quantum dots, one ofwhich carries an electron transport, with the other one onlysupporting fluctuations by being coupled to only one lead. Ifthe capacitive coupling is strong, the occupation of the lateracts as a switch by preventing a second electron to tunnel intothe conducting dot.

sistors70,71.We require that one of them is connected to the left

and right leads and supports the charge and heat current,whereas the other one is only connected to the left lead.The occupation of the latest dot blocks the current on theformer one and therefore works as a switch. Similar pro-cesses can be found in single quantum dots with peculiartunneling couplings30. A related geometry (also with upto one electron) but in a three-terminal configuration hasbeen proposed as a thermal transistor28,72 and realizedexperimentally in metallic Coulomb-blockade islands73.The advantage of this system is that it is enough that

the switch dot is coupled to only one terminal to have thenecessary left-right asymmetry. The conducting quan-tum dot can in principle be totally symmetric. The sepa-ration of the conducting and switching states in differentquantum dots allows them to be tuned independently.Also, this mechanism does not rely on interference andis hence robust against decoherence and noise sources.Let us consider spinless electrons, so the rate equa-

tions can be obtained as a particular case of the two-stateconfiguration, cf. Eq. (9), particularized to the case:ΓR2 = 0. In this case, the states X = 1, 2 denote theoccupation of the conducting and switch quantum dots,respectively.The current through the system can be easily obtained,

and written in a simple form as:

JCQD = J1(ε1)(1 − ρ22), (24)

−10

−5

0

5

10

−10 −5 0 5 10

−10

−5

0

5

10

0

0.4

0

1

-10 0 10

ε 2/kBT

ε1/kBT

0

0.5

1(c)

R

ε 2/kBT

0

0.4(a)

J+[ΓkBT ]

J±

[ΓkBT]

J+

J−

(b)

R

ε2/kBT

(d)

FIG. 12. Rectification of a system of capacitively quantumdots, one of which is tunneled coupled to one lead, only. (a)Forward heat current for ∆T = T , V = 0, Γli = 0.2kBT ,except for ΓR2 = 0, with Γ = ΓLΓR/ΓΣ. (b) Forward andbackward currents along the maximum at ε1 = −0.36kBT .(d) Cut along the same condition of the rectification coeffi-cient plotted in (c) for different level positions. When ε2 < 0,the occupation of the coupled dot suppresses the backwardcurrent, resulting in high rectification coefficients.

in terms of the current through a single channel writ-ten in Eq. (8). Remarkably, the current is conditionedto the switch dot not being occupied. The steady-stateoccupation of the latter:

ρ22 =1− f(ε2, TL)

1 + (Γ+Σ1/ΓΣ1)[1− f(ε2, TL)]

(25)

does not depend on the rate of the switching process,which is therefore determined by the state of the con-ductor dot, and the position of the level ε2 with respectto the chemical potential of the left lead.The cancellation of transport due to the occupation

of the second quantum dot can be observed in Fig. 12.The double peak in the forward heat current vanishesas the energy ε2 becomes negative. The switch dot isthen occupied by an electron, which avoids transitionsthrough the conductor. This effect is most effective whenthe left lead is cold (i.e. in the backward configuration),because the transition to a state where ρ22 → 1 is morepronounced, cf. Fig. 12(b), following the dependence inthe Fermi function (25). The blockade of the backwardcurrent depends exponentially on ε2, so the rectificationcoefficient rapidly increases in the region −kB(T+∆T ) <ε2 < −kBT [see Figs. 12(c) and 12(d)], where J+ is notmuch affected.As all transported electrons have a well defined energy,

ε1, the system does not rectify in the open-circuit config-uration, recovering the behaviour of a single-state: sincecharge and heat currents are proportional to each other,thermal currents vanish at the thermovoltage.

9

Here we have considered a system of two single-electronquantum dots, whose currents are small and hard to de-tect. However, the same mechanism is in principle appli-cable to systems that support larger currents (e.g. quan-tum wires) and are strongly coupled to a switch, openingthe way for the definition of thermal diodes which rectifyconsiderably big currents. The switching process can bedue to the Coulomb interaction with charges in a quan-tum dot, as considered here, or due to internal selectionrules, e.g. spin blockade in double quantum dots74.

VII. CONCLUSIONS

We have investigated the thermal rectification of di-verse quantum dot systems in the Coulomb blockaderegime. Single-electron currents are expected to be small.However, the transport characteristics of these systemscan easily be controlled and scaled up to account forgreater currents. We also identify different mechanismsthat promote rectification and which can in principle betranslated to other systems.A basic ingredient of our results is the strong Coulomb

interaction which introduces correlations between the dif-ferent conduction channels, although other interactionscan also enable the rectification. The details of everyconfiguration determine how the different channels cou-ple to the left and right reservoirs, introducing the nec-essary asymmetries.For a single quantum dot, the presence of two accessi-

ble states with broken mirror symmetry is enough to finda finite rectification, even if the two states are degener-ate. This can be due, for example, to the spin degreeof freedom, in which case the degeneracy can be liftedby means of a magnetic field. Notably, this finding intro-duces a way to enhance the rectification simply by scalingup the number of interacting channels that contribute tothe current. We discuss this possibility by considering asystem of several quantum dots coupled in parallel to thetwo leads.The asymmetry of the tunneling rates can be addition-

ally controlled in a double quantum dot. Hybridization ofthe localized states due to coherent tunneling introducesenergy-resolved and left-right asymmetric rates that canbe tuned by means of gate voltages applied to each quan-tum dot. As a consequence, larger rectification effectsare found, remarkably even approaching R ≈ 1. Unfor-tunately, the huge rectification coefficients correspond toconfigurations with very small thermal currents.Considering that the conducting channels interact with

an energy level that is only coupled to one of the leads,the occupation of this level will act as a switch. This way,the mirror symmetry of the conducting channels is bro-ken, and the current strongly depends on the switch statebeing below the chemical potential of its lead. The block-ade is lifted by thermal fluctuations, which introducesa temperature-dependent threshold. This introduces ahuge rectification effect as the presence of a current re-lies on whether the switch is coupled to the hot or to thecold terminal.

We use this effect in two different configurations: Ina triangular triple quantum dot, tunneling interferenceslead to the formation of a transport dark state in the twoleftmost quantum dots, which avoids tunneling to theright one. An exponential suppression of the backwardcurrent is found for configurations where the forward oneshows a resonance.In a system of capacitively-coupled quantum dots, one

of them serves as a conductor, while the other one istunnel-coupled to one lead, only. The rectification coef-ficient can in this case be controlled with a single gatevoltage coupled to the single-terminal quantum dot. Thisconfiguration is of experimental relevance53,54,56,65–69,73

and can readily be tested.For typical experimental conditions in semiconductor

quantum dots with tunneling rates Γ ∼ 10 GHz andT ∼ 100 mK, heat currents would be of the order of1 fW, well within present day resolution9. In quantumdots defined in two dimensional electron gases, the regimeof application of our results is restricted to low temper-atures (of the order of 0.1–1 K) where Coulomb block-ade effects have been observed. The application of largetemperature gradients is an issue in two dimensional elec-tron gases, but some advances in quantum dots embed-ded into nanowires has been recently achieved36. Also,recent room temperature detection of Coulomb blockadein nanoparticles75, and quantum inteference in molecularjunctions76–80 are promising advances toward the appli-cation of the effects discussed here in thermal devices.Here we have restricted ourselves to the weak-coupling

regime. Exploring these interacting effects in full coher-ent transport and accounting for the effect of possiblesources of dephasing48 remain as issues for future work.

ACKNOWLEDGMENTS

We enjoyed discussions with Fernando Gonzalez-Zalba,Milena Grifoni, Clive Emary, Daniel Manzano, DavidSanchez, and thank Holger Thierschmann for usefulcomments on the manuscript. We acknowledge finan-cial support from the Spanish MINECO via grants No.FIS2015-74472-JIN (AEI/FEDER/UE), No. MTM2017-84446-C2-2-R and No. MTM2014-56948-C2-2-P, and theRamon y Cajal program RYC-2016-20778. M.R.G. alsoacknowledges support fromMECD through the FPU pro-gram.

Appendix A: Single non-degenerate level model

Let us consider the well known model of a quantumdot with a single non-degenerate level at energy ε1. It isusually used when the spin degree of freedom does notplay any role. The two states of the system are definedby whether it is empty or occupied: |X〉 = |0〉, |1〉. Therate equation simply reads:

ρ11 = Γ+Σ1ρ00 − Γ−

Σ1ρ11, (A1)

with Γ±Σ1 = Γ±

L1 +Γ±R1. In the stationary limit, it is easy

to obtain the steady state occupation: ρ00 = Γ−Σ1/(Γ

+Σ1+

10

Γ−Σ1), and ρ11 = Γ+

Σ1/(Γ+Σ1 + Γ−

Σ1). The denominatorwarranties the conservation of probability, 1 = ρ00 + ρ11.It is independent of the lead temperature, as Γ+

l1 +Γ−l1 =

Γl1. With these, it is immediate to obtain the chargecurrent:

I1(ε) = eΓL1ΓR1

ΓL1+ΓR1

[f(ε, TL)− f(ε, TR)]. (A2)

The heat current is tightly coupled: J1/I1 = ε1/e, re-sulting in Eq. (8).

The currents in Eqs. (A2) and (8) only depend ontemperature through the difference of Fermi functions.Hence, exchanging them results in a global change ofsign, i.e. there is no rectification in short-circuit evenwhen the tunneling rates (ΓR1 and ΓL1) were not equal.Note that in the case of asymmetric tunneling rates thestate occupation, ρ11, does change when the temperaturedifference is reversed. This effect, combined with inter-action between levels is what enables rectification in Sec.III.

The tight-coupling relation also avoids rectification inopen-circuit, as it involves I1 = 0.

Appendix B: Many single non-degenerate level

model

The argument in App. A holds if one has several copiesof the same system which do not interact to each other:

J =∑

i

Ji =∑

i

εiΓLiΓRi

ΓLi+ΓRi[f(εi, TL)−f(εi, TR)]. (B1)

The prefactor in the previous expression depends onlyon the couplings of each channel, and is temperature-independent, again resulting in R = 0 for the short-circuit configuration.

The open circuit is in this case different: if not all chan-nels have the same energy, tight charge-energy couplingdoes not hold and therefore rectification is finite.

Appendix C: Quantum dot in open circuit

We consider the two-state quantum dot discussed inSec. III under the open-circuit conditions.The heat current characteristics are strongly affected,

as shown in Fig. 13. First of all, there is no heat flowwhatsoever for the condition ∆E = 0: since the twostates have the same energy, charge and heat currentsbecome proportional to each other,

J2,o−c =

(

ε

e+

V

2

)

I2,o−c. (C1)

This is the so-called tight-coupling limit. It follows triv-ially that the heat current will vanish as well.The tight coupling is lifted under a finite level splitting,

∆E 6= 0, e.g. again due to an applied magnetic field.The cancellation of the charge current does no longerimply that heat flows vanish. Indeed, heat unavoidablyflows from the hot to the cold terminal, showing a singlepeak structure confined in the region where ε2 > 0 andε1 < 0. For positive energies, ε2 > ε1 > 0, the upper levelis rarely occupied, so the system behaves a single levelshowing no current in open-circuit. When both levels arenegative, the rectification increases linearly. A compro-mise between large rectification and non-vanishing cur-rents is found then for ε2 ≈ 0 and ∆E ≈ 3kBT .

0

5

10

−10 −5 0 5 100

5

10

−10 −5 0 5 10

∆E/kBT

ε/kBT

0

0.3(b)

R

∆E/kBT

ε/kBT

0

0.1(a)

J+[ΓkBT ]

FIG. 13. Rectification of a two-state quantum dot in open-circuit. (a) Heat current and (b) rectification coefficient asfunctions of the position of the level and an applied magneticfield. The same parameters as in Fig. 3 are considered.

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